Variational Method Research Articles

Numerical approximations of groundwater flow problem using fractional variational iteration method with fractional derivative of singular and nonsingular kernels

This study aims to present the numerical approximations of groundwater flow problem, namely the time-fractional Boussinesq equation using the fractional variational iteration method. Fractional derivatives in time variable are taken in Caputo and Caputo–Fabrizio sense. This work compares the solutions of the time-fractional Boussinesq equation for the Caputo and Caputo–Fabrizio fractional operators. Graphical representation of the water table head’s behavior for different time values is shown.

Geometric nonlinear vibration theory of the Vierendeel Sandwich Plate based on generalized variational method

Geometric nonlinear vibration theory of the Vierendeel Sandwich Plate based on generalized variational method

A Flexible, Multi-Instrument Optimal Estimation Retrieval for Wind Profiles

Abstract Instruments such as Doppler lidars, radar wind profilers and uncrewed aircraft systems could be used in observation networks to fill in the temporal and spatial gap that exists for low level wind observations. These instruments, however, do not directly observe the wind and require a retrieval to be used to obtain wind estimates from their observations. Also, the depth and uncertainty of observations collected by these instruments varies depending on the environment that they are sampling. Optimal estimation is a variational retrieval method that combines information from a prior data set and observations to retrieve an atmospheric state. This technique can be beneficial to use when observations have large uncertainties or provide insufficient information to obtain the atmospheric state by themselves. A new optimal estimation retrieval for obtaining wind profiles from typical lower atmosphere wind profiling instrumentation has been developed. This retrieval allows for more observations from wind profiling instrumentation to be used when retrieving wind profiles, increases the depth of retrieved profiles, and eliminates vertical data gaps. This retrieval can also be used to easily combine observations from different instruments or even with model data to create combined data wind retrievals that leverage the strengths of the different data sources to retrieve a wind profile that is superior to those obtained by the individual observations or data sources. It is envisioned that this retrieval will be continued to be developed and maintained as community software as lower atmosphere wind observing capabilities further develop and expand.

Existence and multiplicity of solutions for a Steklov eigenvalue problem involving the p(x)-Laplacian-like operator

Using the variational method, we prove theexistence and multiplicity of solutions for a Steklov problem involving the $p(x)$-Laplacian-like operator, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained.

Analytical solution to boundary layer flow and convective heat transfer for low Prandtl number fluids under the magnetic field effect over a flat plate

AbstractThe present study aims to quantify the flow field, flow velocity, and heat transfer features over a horizontal flat plate under the influence of an applied magnetic field, with a particular emphasis on low Prandtl number fluids. Nonlinear partial differential expressions can be incorporated into the ordinary differential framework with the use of appropriate transformations. This research utilizes the variational iteration method (VIM) to approximate solutions for the system of nonlinear differential equations that define the problem. The objective is to demonstrate superior flexibility and broader application of the VIM in addressing heat transfer issues, compared to alternative approaches. The results obtained from the VIM are compared with numerical solutions, revealing a significant level of accuracy in the approximation. The numerical findings strongly suggest that the VIM is effective in providing precise numerical solutions for nonlinear differential equations. The analysis includes an examination of the flow field, velocity, and temperature distribution across various parameters. The study found that improving temperature patterns, velocity distribution, and flow dynamics were all positively impacted by increasing the Prandtl numbers. As a result, this leads to the thickness of the boundary layer to decrease and improves heat transfer at the moving surface. Thus, the convection process becomes more efficient. When the strength of the magnetic field is increased, the velocity of the fluid decreases. This observation aligns with expectations since the magnetic field hampers the natural flow of convection. Notably, the convection process can be precisely controlled by carefully applying magnetic force.

Prediction through quantum dynamics simulations: Photo-excited cyclobutanone.

Quantum dynamics simulations are becoming a standard tool for simulating photo-excited molecular systems involving a manifold of coupled states, known as non-adiabatic dynamics. While these simulations have had many successes in explaining experiments and giving details of non-adiabatic transitions, the question remains as to their predictive power. In this work, we present a set of quantum dynamics simulations on cyclobutanone using both grid-based multi-configuration time-dependent Hartree and direct dynamics variational multi-configuration Gaussian methods. The former used a parameterized vibronic coupling model Hamiltonian, and the latter generated the potential energy surfaces on the fly. The results give a picture of the non-adiabatic behavior of this molecule and were used to calculate the signal from a gas-phase ultrafast electron diffraction (GUED) experiment. Corresponding experimental results will be obtained and presented at a later stage for comparison to test the predictive power of the methods. The results show that over the first 500fs after photo-excitation to the S2 state, cyclobutanone relaxes quickly to the S1 state, but only a small population relaxes further to the S0 state. No significant transfer of population to the triplet manifold is found. It is predicted that the GUED experiments over this time scale will see signals related mostly to the C-O stretch motion and elongation of the molecular ring along the C-C-O axis.

Non-adiabatic direct quantum dynamics using force fields: Toward solvation.

Quantum dynamics simulations are becoming a powerful tool for understanding photo-excited molecules. Their poor scaling, however, means that it is hard to study molecules with more than a few atoms accurately, and a major challenge at the moment is the inclusion of the molecular environment. Here, we present a proof of principle for a way to break the two bottlenecks preventing large but accurate simulations. First, the problem of providing the potential energy surfaces for a general system is addressed by parameterizing a standard force field to reproduce the potential surfaces of the molecule's excited-states, including the all-important vibronic coupling. While not shown here, this would trivially enable the use of an explicit solvent. Second, to help the scaling of the nuclear dynamics propagation, a hierarchy of approximations is introduced to the variational multi-configurational Gaussian method that retains the variational quantum wavepacket description of the key quantum degrees of freedom and uses classical trajectories for the remaining in a quantum mechanics/molecular mechanics like approach. The method is referred to as force field quantum dynamics (FF-QD), and a two-state ππ*/nπ* model of uracil, excited to its lowest bright ππ* state, is used as a test case.

Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities

Abstract In this article, we consider the existence of solutions for nonlinear elliptic equations of the form − Δ u + V ( ∣ x ∣ ) u = Q ( ∣ x ∣ ) f ( u ) , x ∈ R 2 , -\Delta u+V\left(| x| )u=Q\left(| x| )f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{2}, where the nonlinear term f ( s ) f\left(s) has critical exponential growth which behaves like e α s 2 {e}^{\alpha {s}^{2}} , the radial potentials V , Q : R + → R V,Q:{{\mathbb{R}}}^{+}\to {\mathbb{R}} are unbounded, singular at the origin or decaying to zero at infinity. By combining the variational methods, Trudinger-Moser inequality, and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a Mountain-pass-type solution for the above problem under some weak assumptions.

Resonant soliton molecules, asymmetric solitons and the other diverse wave solutions to the (3 + 1)-dimensional generalized Kudryashov-Sinelshchikov equation for liquid with gas bubbles

Under the present study, we focus on developing some exact solutions of the (3 + 1)-dimensional generalized Kudryashov-Sinelshchikov equation (KSE) for the liquid with gas bubbles. First, the resonant soliton molecules on the different planes are extracted via imposing the velocity resonance condition on the N-soliton solutions that constructed via the Hirota bilinear method. Besides, the asymmetric solitons are also derived by choosing the appropriate parameters. In the end, the diverse kinds of travelling wave solutions including the bright wave, dark wave, singular wave and periodic wave solutions that expressed by the trigonometric, hyperbolic and rational functions are constructed by employing three effective approaches, namely the variational method, Sub-equation method and Wang’s direct mapping method-II. The profiles of the attained solutions are depicted to present the dynamical performances. As far as we all know, the results of this research are all new and can enable us figure out the physical properties of the (3 + 1)-dimensional generalized KSE better.

Existence of solutions for elliptic systems involving the fractional p(x)-Laplacian

In this paper, we consider a class of non-homogeneous fractional (p (.), q (.))−Laplacian systems. By using variational methods and combining with the theory of Lebesgue and fractional Sobolev spaces with variable exponents, we prove the existence of solutions to the system.

A 3-D minimum-enstrophy vortex in stratified quasi-geostrophic flows

Applying a variational analysis, a minimum-enstrophy vortex in three-dimensional (3-D) fluids with continuous stratification is found, under the quasi-geostrophic hypothesis. The buoyancy frequency is held constant. This vortex is an ideal limiting state in a flow with an enstrophy decay while energy and generalized angular momentum remain fixed. The variational method used to obtain two-dimensional (2-D) minimum-enstrophy vortices is applied here to 3-D integral quantities. The solution from the first-order variation is expanded on a basis of orthogonal spherical Bessel functions. By computing second-order variations, the solution is found to be a true minimum in enstrophy. This solution is weakly unstable when inserted in a numerical code of the quasi-geostrophic equations. After a stage of linear instability, nonlinear wave interaction leads to the reorganization of this vortex into a tripolar vortex. Further work will relate our solution with maximal entropy 3-D vortices.

Excitation and manipulation of recurring rogue waves via nonlinear compression

We theoretically propose a feasible scheme to generate high-density waves and manipulate them quantitatively. We show that a wave with greater density compression called “recurring rogue wave” can be excited by first generating a rogue wave via quenching and Gaussian space modulation techniques followed by employing a nonlinear compression method. The formation of recurring rogue wave is verified to be a consequence of the combined interaction of nonlinear compression and “self compression”. We find that the maximal density of recurring rogue wave exhibits a nonmonotonic dependence on compression time, which allows us to manipulate recurring rogue waves quantitatively through changing compression time and modulation parameters within a certain range. The analytical results based on variational method qualitatively support the numerical simulation results. Our study provides potential strategies for efficiently generating high-density waves in Bose–Einstein condensate experiments.

Ground state solutions for generalized quasilinear Schrödinger equations

In this paper we consider the generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( x , u ) , x ∈ R N , where V and h are periodic in x i , 1 ⩽ i ⩽ N. By using variational methods, we prove the existence of ground state solutions, i.e., nontrivial solutions with least possible energy.

Dynamics of bouncing convex body

We consider a dynamical system that describes a convex planer body with a smooth boundary, which falls under the influence of gravity and is elastically reflected upon hitting a wall. Under suitable restriction on the system’s parameters, the system is equivalent to a billiard problem within a cylinder. We prove that the dynamics of the system is intimately linked to a variational problem, corresponding to a twist map. Moreover, we utilise the established variational method to investigate the stability of trivial periodic orbits and near-integrable dynamics when this planar body closely approximates a disc with its centre of mass at the geometric centre. Numerical results illustrates our mathematical findings and highlight the system’s intricate dynamics.

A note on Kirchhoff type boundary value problem involving Riemann-Liouville fractional derivative

In this paper, we study some nonlinear Kirchhoff boundary value problems of fractional differential equations involving Riemann Liouville operator. Under appropriate assumptions on the functions in the given problem, we establish the existence of solutions using variational methods combined with the mountain pass theorem. Moreover, an illustrative example is presented to prove the validity of the main result.

Solving the general form of the fractional Black–Scholes with two assets through Reconstruction Variational Iteration Method

Solving the general form of the fractional Black–Scholes with two assets through Reconstruction Variational Iteration Method

AVBAE-MODFR: A novel deep learning framework of embedding and feature selection on multi-omics data for pan-cancer classification

Integration analysis of cancer multi-omics data for pan-cancer classification has the potential for clinical applications in various aspects such as tumor diagnosis, analyzing clinically significant features, and providing precision medicine. In these applications, the embedding and feature selection on high-dimensional multi-omics data is clinically necessary. Recently, deep learning algorithms become the most promising cancer multi-omic integration analysis methods, due to the powerful capability of capturing nonlinear relationships. Developing effective deep learning architectures for cancer multi-omics embedding and feature selection remains a challenge for researchers in view of high dimensionality and heterogeneity. In this paper, we propose a novel two-phase deep learning model named AVBAE-MODFR for pan-cancer classification. AVBAE-MODFR achieves embedding by a multi2multi autoencoder based on the adversarial variational Bayes method and further performs feature selection utilizing a dual-net-based feature ranking method. AVBAE-MODFR utilizes AVBAE to pre-train the network parameters, which improves the classification performance and enhances feature ranking stability in MODFR. Firstly, AVBAE learns high-quality representation among multiple omics features for unsupervised pan-cancer classification. We design an efficient discriminator architecture to distinguish the latent distributions for updating forward variational parameters. Secondly, we propose MODFR to simultaneously evaluate multi-omics feature importance for feature selection by training a designed multi2one selector network, where the efficient evaluation approach based on the average gradient of random mask subsets can avoid bias caused by input feature drift. We conduct experiments on the TCGA pan-cancer dataset and compare it with four state-of-the-art methods for each phase. The results show the superiority of AVBAE-MODFR over SOTA methods.

Transforms-Based Bayesian Tensor Completion Method for Network Traffic Measurement Data Recovery

Network traffic measurement is regarded as the bedrock of next-generation network systems. Its purpose is to monitor the network traffic and provide data support for traffic engineering. For this reason, monitoring traffic data from a network-wide perspective is particularly important. However, the proliferation of network services has led to the explosive growth of network traffic, which has brought significant challenges to the measure of network-wide traffic. Therefore, how to infer network-wide traffic from partial traffic data is extremely important. In this article, a transforms-based Bayesian tensor completion (TBTC) method is proposed to infer network traffic data. First, the heterogeneous network traffic data with missing entries are organized into observation tensors according to temporal dimensions and other attributes. Second, the sparse hierarchical prior is used to induce lateral slices sparsity of factor tensors, which makes the tubal rank of the observation tensor can be estimated. Further, a variational Bayesian inference method is developed for model learning, and an efficient updating method is presented. Finally, two cases of the linear transforms-based tensor completion model are implemented in the experiments. Experimental results on two real-world network traffic datasets validate that the proposed method can efficiently and accurately recover network traffic data.

Scalable semidefinite programming approach to variational embedding for quantum many-body problems

In quantum embedding theories, a quantum many-body system is divided into localized clusters of sites which are treated with an accurate ‘high-level’ theory and glued together self-consistently by a less accurate ‘low-level’ theory at the global scale. The recently introduced variational embedding approach for quantum many-body problems combines the insights of semidefinite relaxation and quantum embedding theory to provide a lower bound on the ground-state energy that improves as the cluster size is increased. The variational embedding method is formulated as a semidefinite program (SDP), which can suffer from poor computational scaling when treated with black-box solvers. We exploit the interpretation of this SDP as an embedding method to develop an algorithm which alternates parallelizable local updates of the high-level quantities with updates that enforce the low-level global constraints. Moreover, we show how translation invariance in lattice systems can be exploited to reduce the complexity of projecting a key matrix to the positive semidefinite cone.

Federated Variational Generative Learning for Heterogeneous Data in Distributed Environments

Distributedly training models across diverse clients with heterogeneous data samples can significantly impact the convergence of federated learning. Various novel federated learning methods address these challenges but often require significant communication resources and local computational capacity, leading to reduced global inference accuracy in scenarios with imbalanced label data distribution and quantity skew. To tackle these challenges, we propose FedVGL, a Federated Variational Generative Learning method that directly trains a local generative model to learn the distribution of local features and improve global target model inference accuracy during aggregation, particularly under conditions of severe data heterogeneity. FedVGL facilitates distributed learning by sharing generators and latent vectors with the global server, aiding in global target model training from mapping local data distribution to the variational latent space for feature reconstruction. Additionally, FedVGL implements anonymization and encryption techniques to bolster privacy during generative model transmission and aggregation. In comparison to vanilla federated learning, FedVGL minimizes communication overhead, demonstrating superior accuracy even with minimal communication rounds. It effectively mitigates model drift in scenarios with heterogeneous data, delivering improved target model training outcomes. Empirical results establish FedVGL's superiority over baseline federated learning methods under severe label imbalance and data skew condition. In a Label-based Dirichlet Distribution setting with α=0.01 and 10 clients using the MNIST dataset, FedVGL achieved an exceptional accuracy over 97% with the VGG-9 target model.